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Friday, July 5, 2013

Optimal Pricing for a Simple Monopolist

# A single price monopolist is a monopolist because it is the only supplier of a particular product. The monopolist therefore has the power to choose a price to sell the product at.# Those who have a willingness to pay which is greater than the price will buy the good while those who have a willingness to pay for the good which is less than the chosen price will not but it.# Our monopolist is a broadband internet supplier within a city.# For now let's say they only offer one bundle.# Let's generate our consumers
npeep <- 2000# Number of potential consumers
wtp <- 45 + rnorm(npeep)*15# Each person has a different willingness to pay which# To figure out the demand curve we count the number of people willing to pay at least as much as the offering price.
maxop <- 90# Max offering price
op <- 0:maxop # Offering price ranges from 0 to maxop
qd <- rep(NA,length(op))# Quantity demandedfor(i in1:length(op)) qd[i] <- sum(wtp>=op[i])
mc <- qd*.01# Marginal cost is increasing though this is not a neccessity# For something like broadband services we might think that up to a point marginal costs might be decreasing since the cost of adding one more customer might be less than the cost of adding the previous customer.plot(qd, op, type="l", xlab="Quantity", ylab="Price, Marginal Cost - Red",
main="Demand for Broadband Internet", lwd=2)abline(h=0, lwd=2)lines(qd, mc,col="red", lwd=2)

# The monopolist must choose a price in which to sell services at.# If the monopolist chooses mc=p then the monopolist will not make any money but the consumers will be very happy.# We know that the optimal point for the monopolist is at the point where marginal revenue curve intersects the marginal cost curve.# Let's see if we can find it.
tr <- tp <- tc <- rep(NA,length(op))# Total revenue, total profit, total cost vectors# Calculate total cost
qd.gain <- qd[-length(qd)]-qd[-1]
qd.gain[length(qd.gain)+1] <- qd.gain[length(qd.gain)]for(i in1:length(op)) tc[i] <- sum((mc*qd.gain)[length(qd):i])
tr <- qd*op
tp <- tr-tc
minmax <- function(...)c(min(...),max(...))plot(minmax(op),minmax(tr,tp), type="n", ylab="Total Revenue - Blue, Total Profit - Red",
xlab="Price", main="We can see optimal pricing\nfor the monopolist is around 39 dollars")grid()abline(h=0, lwd=2)abline(v=39,col="red", lwd=2)lines(op,tr,col="blue", lwd=3)lines(op,tp,col="red", lwd=2)

# We can see at the price around 18 which would be the optimal price for the consumer, the supplier is making almost no profits.# The last thing we might wish to consider to Total Surplus or total system efficiency which is defined as that which the consumer benefits by purchasing a good below the consumers willingness to pay plus that of the suppliers profit at that price.
cs <- tr
for(i in1:length(op)) cs[i] <- sum((wtp[wtp>=op[i]]-op[i]))
tts <- cs+tp
op[tts==max(tts)]# Check the optimatal societal priceplot(c(min(op),max(op)),c(min(cs,tp),max(cs,tp)), type="n",
main="Optimal societal pricing is at\n mc=wtp which is $19",
xlab="Price",
ylab="purple=CS, blue=PS, black=TS")lines(op, cs,col="purple", lwd=2)lines(op, tp,col="blue", lwd=2)lines(op, tts, lwd=2)abline(h=0,col="red", lwd=2)

1 comment
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